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Math Help - Calculus Circle Word Problem

  1. #1
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    Calculus Circle Word Problem

    These word problems are driving me crazy.....

    Clyde and CeCe are running around a circular track of radius 50 meters in the same direction. Clyde is running at a rate of 1.3 m/sec and CeCe is running at a rate of 0.9 m/sec. When Clyde is behind CeCe as pictured and the (straight) distance between them is 82 meters, how is the (straight) distance between them changing?
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  2. #2
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    Let's try using the formula for a chord and the formula s=r{\theta}

    chord length is given by L=2rsin(\frac{\theta}{2})

    Arc length is given by s=r{\theta}

    Clyde is catching up to CeCe, so their distance along the arc is decreasing at -0.4 m/sec.

    That is ds/dt = -.4.

    \frac{ds}{dt}=r\frac{d{\theta}}{dt}+{\theta}\frac{  dr}{dt}

    -.4=50\frac{d{\theta}}{dt}+0. dr/dt is 0 because the radius does not change.

    \frac{d{\theta}}{dt} = -.008

    When they are 82 feet apart along the chord, then using 2rsin(\frac{\theta}{2})

    {\theta}=2sin^{-1}(\frac{82}{100})=1.92282 \;\ rad.

    So, let's differentiate implicitly our chord formula.

    \frac{dL}{dt}=50cos(\frac{\theta}{2})\frac{d{\thet  a}}{dt}

    \frac{dL}{dt}=50cos(\frac{1.92282}{2})(-.008)=-.229

    The distance between them is decreasing at .229 m/sec.

    Let's see if anyone concurs. Check my work.
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